Problem: Multiply and simplify the following complex numbers: $({5+2i}) \cdot ({-5-i})$
Solution: Complex numbers are multiplied like any two binomials. First use the distributive property: $ ({5+2i}) \cdot ({-5-i}) = $ $ ({5} \cdot {-5}) + ({5} \cdot {-i}) + ({2i} \cdot {-5}) + ({2i} \cdot {-i}) $ Then simplify the terms: $ (-25) + (-5i) + (-10i) + (-2i^2) $ Imaginary unit multiples can be grouped together. $ -25 + (-5 - 10)i - 2 i^2 $ After we plug in $i^2 = -1$, the result becomes $ -25 + (-5 - 10)i - (-2) $ The result is simplified: $ (-25 + 2) + (-15i) = -23-15i $